Each year at our school we have one (or two) teachers who direct a seminar to colleagues–everyone gets a small stipend–directors and seminarians (as we call them). This past year we worked on topics in mathematics from number theory to probability and from fractals to the concept of infinity. It was so cool.
While finishing my next blog posts (graphic novels, 2nd person narration [!], teaching poetry, and much more) I have posted my paper from this past summer below.
Confessions of a Wannabe Math Teacher
By Daniel McMahon
Chapter 1—Wherin are made Proleptic Remarks
My parents tell the story that I was sent for a nap as child and, to keep me there, I was told to start counting, some short time later I came down and announced I had counted to infinity. When I was in second grade my love affair with baseball and matters statistical lead my father to teach me how to calculate batting averages with a slide rule. In those days each box score did not provide a batting average and you had to wait until Sundays to read the complete listings of all averages in the major leagues. I was always a good student in math when I liked the teacher—and a poor one when I didn’t—a certain mark of the immature student at any age. But my fascination with numbers has always been just below the surface throughout my life despite my major and career and my study this past summer has given me a chance to indulge this fascination in several ways.
Here are two brief stories about the summer of 2012: I think I can prove that any whole number can be achieved through the sum of Fibonacci numbers. You can also achieve any whole number through the sum of only odd numbers (but this cannot be done with even numbers)—of course there are many more odd numbers than Fibonacci numbers. But how about this—can all whole numbers be achieved by the sum of only prime numbers? At first it appears that you can since among the first the first 100 numbers there are 26 prime numbers (and 49 odd numbers) (there are only 10 Fibonacci numbers in the first 100 and the 26th Fibonacci number is 196,418; 97 is the 26th prime number). But eventually prime numbers get further and further apart so I don’t know the answer.
A second story from this summer comes from when I was kneeling in the Church of the Holy Sepulchre in Jerusalem at 6:00 am mass on July 16. I was staring at the floor and the remarkable tile shapes that make the floor. Here are two of them
How cool are these shapes using triangles and squares (and other four-sided figures) to make pentagons and octagons (lots of 3s, 5s and 8s—all Fibonacci numbers here)?
Chapter 2—Wherein the Main Argument is Presented With Several Digressions
Pierre de Fermat (1601-1665) was a brilliant mathematician who collaborated with Pascal on number theory and probability, argued with Descartes through letters that were often refereed by his friend Marin Mersenne (he of the eponymous “Mersenne primes”—primes derived from 2n-1). Fermat is now known almost exclusively for his “Last Theorem” which was that there are no natural numbers x, y, and z such that xn + yn = zn in which n is a natural number greater than 2. Fermat indicated he had the proof but couldn’t fit it in the margins of the book he was writing in—the theory was finally proved in 1994 and 1995 by Andrew Wiles and Richard Taylor.
Fermat was intensely private and published almost nothing in his life time though friends begged him to do so; his mathematic prowess and remarkable insights in physics influenced not only his contemporaries but Euler, Leibniz, Newton, Huygens, and others. Had he published he might have been given credit as the inventor of number theory. One of my favorite insights by Fermat comes from his work on optics and is known as Fermat’s principle or Fermat’s principle of least time. “…When light proceeds by any path from point A to another point B, the time required in its passage is either a minimum or a maximum as compared to other, arbitrarily chosen, adjacent paths. If the light is reflected from A to B by a plane surface or is refracted at a plane surface on its way from A to B, the time is minimum. For a curved reflecting surface, the time is a minimum if the surface has less curvature than the ‘aplanatic’ surface osculating with it at the same point (i.e., the surface which gives rise to no spherical aberration); and this holds true also for a curved refracting surface” (Van Nostrand’s Scientific Encyclopedia, p. 1121) This insight was foundational for the study of reflection and refraction and ran counter the prevailing notion—posited, incorrectly, by Descartes—that “light travels more rapidly in the denser of two media involved in the refraction” (Encyclopedia Britannica Micropaedia,5thedition. 739). Fermat’s insight leads to Snell’s Law and a great deal of optical theory. (See Appendix A)
An example of light “choosing” the fastest time to get from point A to point B is shown below—with two other, slower solutions that light might have “chosen.”
[I cannot get this particular graphic to reproduce–so I’ll describe it. Imagine a horizontal line. Now imagine a line approaching the horizontal line from the top, crossing the horizontal line but changing direction slightly. Now draw two dotted lines, one of which makes a right angle. In the first, bring the dotted line parallel to the horizontal line until, at a right angle it drops and reaches the end point. Next, draw a dotted line from the starting point and following a straight line (through the horizontal line) to the end point.]
There are philosophical implications to Fermat’s insight. It appears that light acts with intentionality. It always takes the shortest amount of time—but it can only do that if it knows where the end point, or telos, (or where you are going to see it) is going to be. Any deviation from the path once it has started would be slower than the fastest possible path. Most people ignore this revelation and indeed it’s not necessary to consider it to solve problems with it.
The mathematician and writer Keith Devlin (also a regular contributor to NPR) has a chapter in his book The Math Instinct called “Elvis: The Welsh Corgi Who Knows Calculus.” Elvis, whose owner Tim Pennings in a math professor, seems to be aware of Fermat’s Principle. While at the beach one day Tim was throwing a ball in the water and Elvis would run part way down the beach (he would not jump right in the water but neither would he run far enough to make a right angle turn) and then swim at an angle to the ball. Well, this looks a lot like Fermat’s principle. Think of a right triangle, ACB where AB is the hypotenuse and AC is the longer of the other two lines, BC the shorter. D is a point along AC. Tim’s throw would be AB but Elvis’ run/swim would be AD then DB.
The next time Tim went to the beach with Elvis:
“He brought the ball, along tape measure, a stopwatch, and his swimsuit. Time after time—for a total of 35 repetitions in all—Tim hurled the ball, started the stopwatch, raced along the beach after his dog, dropped marker where Elvis dived into the water, noted the time it had taken Elvis to reach that point, and then followed him in, splaying the tape measure out behind him. Although Tim was left behind on the beach, he is a strong swimmer, and was generally able to catch up with Elvis before the dog reached the ball, and was able to note the time it took Elvis to swim to the ball. Tim then swam back to the beach, noted the point where he hit land, and then used the tape measure to determine the actual lengths AD and AC. On average, Tim threw the ball 20 meters (65 feet) along the beach and 10 meters (33 feet) into the water.” P. 20
Why does the dog automatically solve this problem? Well, at least one idea would seem that natural selection has given some advantage to animals that exhibit better judgment in solving this problem. It turns out that we all enact versions of the solution to this problem in various ways—chasing a Frisbee or a fly ball we do NOT run in a straight line but in a line that allows us to track the flying object in the most efficient way. Though this exercise helps us understand the practical application of Fermat’s principle it leaves a gap in considering the philosophical suggestions opened by Fermat’s principle, namely the idea of intentionality.
Ted Chiang is a contemporary writer of speculative fiction whose stories I have been teaching for several years now and his “Story of Your Life” examines this exact point and blends it with a fascinating take on the Sapir-Whorf hypothesis in linguistics— “The principle of linguistic relativity holds that the structure of a language affects the ways in which its speakers are able to conceptualize their world, i.e. their world view”( Wikipedia, “Sapir-Whorf Hypothesis”).
A linguist and a physicist (Louise Banks and Gary Donnelly) are teamed together to work with an alien race to communicate with them. The aliens, known as Heptapods (for their 7 appendages on a radial body that could be going frontwards or backwards at any time), have two languages—a spoken language and a written language that does not correspond to the spoken language. The written language appears as a picture—it is non-linear, non-glottographic—i.e. the “writing” does not correspond to a sound—something like the red circle with a strikethrough which indicates no “doing whatever is struck through”—but without sound. If you had a language—a writing system—that was pictgographic, could it reflect knowledge of teleology that was already set. You would know the future—as light seems to do. Louise Banks, the linguist describes it in the following way:
“Was it actually possible to know the future? Not simply to guess at it; was it possible to know what was going to happen, with absolute certainty and in specific detail? Gary [the physicist] once told me that the fundamental laws of physics were time-symmetric, that there was no physical difference between past and future. Given that, some might say, “yes, theoretically.” But speaking more concretely, most would answer “no,” because of free will.
I liked to imagine the objection as a Borgesian fabulation: consider a person standing before the Book of Ages, the chronicle that records every event, past and future. Even though the text has been photoreduced from the full-sized edition, the volume is enormous. With magnifier in hand, she flips through the tissue- thin leaves until she locates the story of her life. She finds the passage that describes her flipping through the Book of Ages, and she skips to the next column, where it details what she’ll be doing later in the day: acting on information she’s read in the Book, she’ll bet $100 on the racehorse Devil May Care and win twenty times that much.
The thought of doing just that had crossed her mind, but being a contrary sort, she now resolves to refrain from betting on the ponies altogether.
There’s the rub. The Book of Ages cannot be wrong; this scenario is based on the premise that a person is given knowledge of the actual future, not of some possible future. If this were Greek myth, circumstances would conspire to make her enact her fate despite her best efforts, but prophecies in myth are notoriously vague; the Book of Ages is quite specific, and there’s no way she can be forced to bet on a racehorse in the manner specified. The result is a contradiction: the Book of Ages must be right, by definition; yet no matter what the Book says she’ll do, she can choose to do otherwise. How can these two facts be reconciled?
They can’t be, was the common answer. A volume like the Book of Ages is a logical impossibility, for the precise reason that its existence would result in the above contradiction. Or, to be generous, some might say that the Book of Ages could exist, as long as it wasn’t accessible to readers: that volume is housed in a special collection, and no one has viewing privileges.
The existence of free will meant that we couldn’t know the future. And we knew free will existed because we had direct experience of it. Volition was an intrinsic part of consciousness.
Or was it? What if the experience of knowing the future changed a person? What if it evoked a sense of urgency, a sense of obligation to act precisely as she knew she would? 162-163″
The resolution in the story is that Louise Banks, by learning to think as the Heptapods do—because she learns their written language—comes to know the future. Since the writing is non-linear and the entire text is embedded in the first stroke of the pen, one has to know what the conclusion of the “written discourse” will be before it is begun. (An interesting detour about language comes from experiments by Lera Boroditsky. “While English says ‘she broke the bowl’ even if it smashed accidentally (she dropped something on it, say), Spanish and Japanese describe the same event more like ‘the bowl broke itself.’ ‘When we show people video of the same event,’ says Boroditsky. ‘English speakers remember who was to blame even in an accident, but Spanish and Japanese speakers remember it less well than they do intentional actions. It raises questions about whether language affects even something as basic as how we construct our ideas of causality.’”)
Because Louise knows the future she must act it out as a performative act—like the actress who must hit her marks and say her line to bring into being what will be. Think of God saying “Let there be light.” It is in the act of saying that things happen—not in the thinking about them that creation happens. There are examples such as “I christen this ship” or “I now pronounce you husband and wife” that are also performative speech acts. So even though from this point of view you “lose” free will you gain the creative responsibility for bringing reality into being.
I find this story a good way to get the students to think about language, knowledge, free will, sub specie aeternitatis (God’s point of view), Fermat, physics, and philosophy. That’s a good math story.
“In optics, Fermat recognized that of all possible paths, light takes the path that takes the least time; this fundamental rule is known as Fermat’s principle” (Columbia Encyclopedia, 6th edition, p. 975). Here is another version of Fermat’s principle with the complementary angles shown.
Works Cited and Consulted
Chiang, Ted. Stories of Your Life and Others. New York: Orb Books, 2002.
The Columbia Viking Encyclopedia, Sixth Edition. New York: Columbia University Press, 2000.
Concise Dictionary of Scientific Biography, Second Edition. New York: Charles Scribner’s Sons, 2000.
Devlin, Keith. The Math Instinct. New York: Basic Books, 2006.
Encyclopedia Brittanica Micropaedia 15th edition Chicago, 1998.
“Fermat’s Principle,” in Wikipedia, http://en.wikipedia.org/wiki/Fermat%27s_principle.
Van Nostrand’s Scientific Encyclopedia, Seventh edition. New York: Van Nostrand Reinhold, 1998.
Young, Robyn, ed. Notable Mathematicians. Detroit: Gale, 1998.